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Theorem cygabl 18484
 Description: A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
cygabl (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)

Proof of Theorem cygabl
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2752 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2752 . . 3 (.g𝐺) = (.g𝐺)
31, 2iscyg3 18480 . 2 (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)))
4 eqidd 2753 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (Base‘𝐺) = (Base‘𝐺))
5 eqidd 2753 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (+g𝐺) = (+g𝐺))
6 simpll 807 . . . 4 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Grp)
7 eqeq1 2756 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝑦 = (𝑛(.g𝐺)𝑥) ↔ 𝑎 = (𝑛(.g𝐺)𝑥)))
87rexbidv 3182 . . . . . . . . 9 (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g𝐺)𝑥)))
9 oveq1 6812 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝑛(.g𝐺)𝑥) = (𝑚(.g𝐺)𝑥))
109eqeq2d 2762 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑎 = (𝑛(.g𝐺)𝑥) ↔ 𝑎 = (𝑚(.g𝐺)𝑥)))
1110cbvrexv 3303 . . . . . . . . 9 (∃𝑛 ∈ ℤ 𝑎 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥))
128, 11syl6bb 276 . . . . . . . 8 (𝑦 = 𝑎 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥)))
1312rspccv 3438 . . . . . . 7 (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) → (𝑎 ∈ (Base‘𝐺) → ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥)))
1413adantl 473 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (𝑎 ∈ (Base‘𝐺) → ∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥)))
15 eqeq1 2756 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑦 = (𝑛(.g𝐺)𝑥) ↔ 𝑏 = (𝑛(.g𝐺)𝑥)))
1615rexbidv 3182 . . . . . . . 8 (𝑦 = 𝑏 → (∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) ↔ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
1716rspccv 3438 . . . . . . 7 (∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥) → (𝑏 ∈ (Base‘𝐺) → ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
1817adantl 473 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → (𝑏 ∈ (Base‘𝐺) → ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
19 reeanv 3237 . . . . . . . 8 (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) ↔ (∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥) ∧ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)))
20 zcn 11566 . . . . . . . . . . . . . 14 (𝑚 ∈ ℤ → 𝑚 ∈ ℂ)
2120ad2antrl 766 . . . . . . . . . . . . 13 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℂ)
22 zcn 11566 . . . . . . . . . . . . . 14 (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
2322ad2antll 767 . . . . . . . . . . . . 13 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℂ)
2421, 23addcomd 10422 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) = (𝑛 + 𝑚))
2524oveq1d 6820 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑛 + 𝑚)(.g𝐺)𝑥))
26 simpll 807 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝐺 ∈ Grp)
27 simprl 811 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑚 ∈ ℤ)
28 simprr 813 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
29 simplr 809 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → 𝑥 ∈ (Base‘𝐺))
30 eqid 2752 . . . . . . . . . . . . 13 (+g𝐺) = (+g𝐺)
311, 2, 30mulgdir 17766 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺))) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
3226, 27, 28, 29, 31syl13anc 1475 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛)(.g𝐺)𝑥) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
331, 2, 30mulgdir 17766 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑥 ∈ (Base‘𝐺))) → ((𝑛 + 𝑚)(.g𝐺)𝑥) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3426, 28, 27, 29, 33syl13anc 1475 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑛 + 𝑚)(.g𝐺)𝑥) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3525, 32, 343eqtr3d 2794 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
36 oveq12 6814 . . . . . . . . . . 11 ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)))
37 oveq12 6814 . . . . . . . . . . . 12 ((𝑏 = (𝑛(.g𝐺)𝑥) ∧ 𝑎 = (𝑚(.g𝐺)𝑥)) → (𝑏(+g𝐺)𝑎) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3837ancoms 468 . . . . . . . . . . 11 ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑏(+g𝐺)𝑎) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥)))
3936, 38eqeq12d 2767 . . . . . . . . . 10 ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → ((𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎) ↔ ((𝑚(.g𝐺)𝑥)(+g𝐺)(𝑛(.g𝐺)𝑥)) = ((𝑛(.g𝐺)𝑥)(+g𝐺)(𝑚(.g𝐺)𝑥))))
4035, 39syl5ibrcom 237 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
4140rexlimdvva 3168 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑎 = (𝑚(.g𝐺)𝑥) ∧ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
4219, 41syl5bir 233 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥) ∧ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
4342adantr 472 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → ((∃𝑚 ∈ ℤ 𝑎 = (𝑚(.g𝐺)𝑥) ∧ ∃𝑛 ∈ ℤ 𝑏 = (𝑛(.g𝐺)𝑥)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
4414, 18, 43syl2and 501 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → ((𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎)))
45443impib 1108 . . . 4 ((((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
464, 5, 6, 45isabld 18398 . . 3 (((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) ∧ ∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Abel)
4746r19.29an 3207 . 2 ((𝐺 ∈ Grp ∧ ∃𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∃𝑛 ∈ ℤ 𝑦 = (𝑛(.g𝐺)𝑥)) → 𝐺 ∈ Abel)
483, 47sylbi 207 1 (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1624   ∈ wcel 2131  ∀wral 3042  ∃wrex 3043  ‘cfv 6041  (class class class)co 6805  ℂcc 10118   + caddc 10123  ℤcz 11561  Basecbs 16051  +gcplusg 16135  Grpcgrp 17615  .gcmg 17733  Abelcabl 18386  CycGrpccyg 18471 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-n0 11477  df-z 11562  df-uz 11872  df-fz 12512  df-seq 12988  df-0g 16296  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-grp 17618  df-minusg 17619  df-mulg 17734  df-cmn 18387  df-abl 18388  df-cyg 18472 This theorem is referenced by:  lt6abl  18488  frgpcyg  20116
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